Este trabalho apresenta uma demonstração completa do Teorema de L. Hörmander sobre a divisão de distribuições (temperadas) por polinômios. O caso n=1 é apresentado detalhadamente e serve como motivação para as técnicas utilizadas no caso geral. Todos os pré-requisitos para a demonstração de Hörmander (os Teoremas de Seidenberg-Tarski, de Puiseux e da Extensão de Whitney) são discutidos com detalhes. Como conseqüência do Teorema, segue que todo operador diferencial parcial linear com coeficientes constantes não nulo admite solução fundamental temperada. / This dissertation presents a thorough proof of L. Hörmander\'s theorem on the division of (tempered) distributions by polynomials. The case n=1 is discussed in detail and serves as a motivation for the techniques that are utilised in the general case. All the prerequisites for Hörmander\'s proof (the Theorems of Seidenberg-Tarski, of Puiseux and Whitney\'s Extension Theorem) are discussed in detail. As a consequence of this theorem, it follows that every non zero partial diffe\\-rencial operator with constant coefficients has a tempered fundamental solution.

Master of Science / Department of Mathematics / Marianne Korten / Distribution theory is an important tool in studying partial differential equations. Distributions are linear functionals that act on a space of smooth test functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. There are different possible choices for the space of test functions, leading to different spaces of distributions. In this report, we take a look at some basic theory of distributions and their Fourier transforms. And we also solve some typical exercises at the end.

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The whole thesis comprises six chapters, where the running theme focuses on the development of statistical methods and distribution theory, with applications to finance. It begins with Chapter 1, which provides the introduction and background to my thesis. This is then followed by Chapters 2 through to 6, which provide the main contributions. The exact distribution of the sum of more than two independent beta random variables is not a known result. Even in terms of approximations, only the normal approximation is known for the sum. Motivated by Murakami (2014), Chapter 2 derives a saddlepoint approximation for the distribution of the sum. An extensive simulation study shows that it always gives better performance than the normal approximation. Jin et al. (2016) proposed a novel moments based approximation based on the gamma distribution for the compound sum of independent and identical random variables, and illustrated their approximation through the use of six examples. Chapter 3 revisits four of their examples, and it is shown that moments based approximations based on simpler distributions can be good competitors. The moments based approximations are also shown to be more accurate than the truncated versions of the exact distribution of the compound sum. Results regarding the performances of the approximations are provided, which could be useful in determining which approximation should be used given a real data set. The estimation of the size of large populations can often be a significant problem. Chapter 4 proposes a new population size estimator and provides a comparison of its performance with two recent estimators known in the literature. The comparison is based on a simulation study and applications to two real big data sets from the Twitter and LiveJournal social networks. The proposed estimator is shown to outperform the known estimators, at least for small sample sizes. In recent years, with a growing interest in big or large datasets, there has been a rise in the application of large graphs and networks to financial big data. Much of this research has focused on the construction and analysis of the network structure of stock markets, based on the relationships between stock prices. Motivated by Boginski et al. (2005), who studied the characteristics of a network structure of the US stock market, Chapter 5 constructs network graphs of the UK stock market using the same method. Four distributions are fitted to the degree density of the vertices from these graphs: the Pareto I, Frechet, lognormal, and generalised Pareto distributions, and their goodness of fits are assessed. Results show that the degree density of the complements of the market graphs, constructed using a negative threshold value close to zero, can be fitted well with the Frechet and lognormal distributions. Chapter 6 analyses statistical properties of the largest cryptocurrencies (determined by market capitalisation), of which Bitcoin is the most prominent example. The analysis characterises their exchange rates versus the US Dollar by fitting parametric distributions to them. It is shown that cryptocurrency returns are clearly non-normal, however, no single distribution fits well jointly to all of the cryptocurrencies analysed. We find that for the most popular cryptocurrencies, such as Bitcoin and Litecoin, the generalised hyperbolic distribution gives the best fit, whilst for the smaller cryptocurrencies the normal inverse Gaussian distribution, generalised t distribution, and Laplace distribution give good fits. The results are important for investment and risk management purposes.

In order to investigate the phenomenon of the distribution of gains from international trade, Arghiri Emmanuel's ideas are firs t critically discussed, particularly in relation to the traditional Ricardian framework as applied to labor-surplus economics. It is found that Emmanuel's concept of unequal exchange, which has been termed non-equivalent exchange by Jan Otto Anderson, has certain theoretical drawbacks. In particular, it has been pointed out that the question involved is not one to prove that the poor countries are actually worse off through trade as suggested by Emmanuel. The question involved is rather one of redistribution of gains from trade as has been voiced in the search for a new international economic order by the members of some developing countries in the U.N. Such an approach leads to the adoption of the concept of a generalized asymmetric exchange as the measure of unequal exchange. This generalization has been achieved in terms of Leontief 's input output analysis. Such a measure coincides with the disjunctive exchange approach when the input-output coefficients are modified over time. The Leontief input-output analysis leads to an aggregation problem which has been solved by taking labor as the only primary factor of production - an approach standardized by Leontief himself. According to this measure, the extent of unequal exchange can be quite different from those obtained by the measures suggested by Emmanuel, Shanin and others. It has been pointed out that there is no a-priori reason to believe that a poor country necessarily gains less than its rich counterpart. Indeed, the test that has been made of the measure in the case of trade between Ecuador and the USA shows that it is Ecuador rather the U. S. which gains more from trade between them. The study also suggests some policy recommendations for reducing unequal exchange with special reference to labor-surplus economics.

Typescript Includes bibliographical references

Thesis (Ph.D.)--University of Adelaide, Dept. of Mathematics, 1965. / Typescript. Includes bibliographical references.

Title: Distribution functions of asteroid physical properties Author: Helena Cibulková Institute: Astronomical Institute of Charles University Supervisor: Mgr. Josef Ďurech, Ph.D., Astronomical Institute of Charles Univer- sity Abstract: In this thesis, I utilize photometric data sparse in time produced by all-sky surveys and investigate physical properties of large asteroid populations. In principle, the individual approach to asteroid modeling cannot compass all objects because new asteroids are continually discovered and we do not have enough data for them. Therefore, in this work I present an essentially different, statistical approach. In a series of papers, we developed two independent methods which use a triaxial-ellipsoid approximation, and we test their applicability and limits. We prove they can be used to the photometric databases like Lowell Observatory database or Pan-STARRS. The output quantities are distributions of the spin axis directions and shape elongations for asteroid populations, and using the Kolmogorov-Smirnov test we search for differences among them. The main result of my work is that the distribution of ecliptical longitudes of spin axes is nonuniform. Moreover, this nonuniformity is more significant for asteroids with low orbital inclinations and the distribution is dependent on...

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The purpose of this dissertation was to empirically investigate the Type I error and power rates of four data transformations that produce a variety of non-normal distributions. Specifically, the transformations investigated were (a) the g-and-h, (b) the generalized lambda distribution (GLD), (c) the power method, and (d) the Burr families of distributions in the context of between-subjects and within-subjects analysis of variance (ANOVA). The traditional parametric F tests and their nonparametric counterparts, the Kruskal-Wallis (KW) and Friedman (FR) tests, were selected to be used in this investigation. The four data transformations produce non-normal distributions that have either valid or invalid probability density functions (PDFs). Specifically, the data generating procedures will produce distributions with valid PDFs if and only if the transformations are strictly increasing - otherwise the distributions are considered to be associated with invalid PDFs. As such, the primary objective of this study was to isolate and investigate the behaviors of the four data transformation procedures themselves while holding all other conditions constant (i.e., sample sizes, effect sizes, correlation levels, skew, kurtosis, random seed numbers, etc. all remain the same). The overall results of the Monte Carlo study generally suggest that when the distributions have valid probability density functions (PDFs) that the Type I error and power rates for the parametric (or nonparametric) tests were similar across all four data transformations. It is noted that there were some dissimilar results when the distributions were very skewed and near their associated boundary conditions for a valid PDF. These dissimilarities were most pronounced in the context of the KW and FR tests. In contrast, when the four transformations produced distributions with invalid PDFs, the Type I error and power rates were more frequently dissimilar for both the parametric F and nonparametric (KW, FR) tests. The dissimilarities were most pronounced when the distributions were skewed and heavy-tailed. For example, in the context of a parametric between subjects design, four groups of data were generated with (a) sample sizes of 10, (b) standardized effect size of 0.50 between groups, (c) skew of 2.5 and kurtosis of 60, (d) power method transformations generating distributions with invalid PDFs, and (e) g-and-h and GLD transformations both generating distributions with valid PDFs. The power results associated with the power method transformation showed that the F-test (KW test) was rejecting at a rate of .32 (.86). On the other hand, the power results associated with both the g-and-h and GLD transformations showed that the F-test (KW test) was rejecting at a rate of approximately .19 (.26). The primary recommendation of this study is that researchers conducting Monte Carlo studies in the context described herein should use data transformation procedures that produce valid PDFs. This recommendation is important to the extent that researchers using transformations that produce invalid PDFs increase the likelihood of limiting their study to the data generating procedure being used i.e. Type I error and power results may be substantially disparate between different procedures. Further, it also recommended that g-and-h, GLD, Burr, and fifth-order power method transformations be used if it is desired to generate distributions with extreme skew and/or heavy-tails whereas third-order polynomials should be avoided in this context.